Experimental analysis of spring hardening and softening nonlinearities in microelectromechanical oscillators. Sarah Johnson Department of Physics, University of Florida Mentored by Dr. Yoonseok Lee Abstract Micro-electro-mechanical systems or MEMS are used in a variety of today s technology and can be modeled using equations for nonlinear damped harmonic oscillators. Mathematical expressions have been formulated to determine resonance frequency shifts as a result of hardening and softening effects in MEMS devices. In this work we experimentally test the previous theoretical analysis of MEMS resonance frequency shifts in the nonlinear regime. Devices are put under low pressure and swept through a range of frequencies with varying AC and DC excitation voltages to detect shifts in the resonant frequency. Mathematical models have predicted greater nonlinearities with higher AC amplitudes and spring softening with high DC voltages. Further research is being conducted to compare MEMS experimental data to earlier theoretical models. Our research will assist in confirming a more accurate model for MEMS devices, which could result in improved micro-technology in products such as gyroscopes, accelerometers, and displays.
I. Introduction Micro-electro-mechanical systems or MEMS have a growing impact on a wide variety of today s technology. Ranging from 1 to 100 microns in size, about the thickness of a human hair, their capabilities for miniaturization and multiplicity make them extremely useful in microelectronics. Cooperative work from many micro-machines is the most efficient way to perform a task. It is cost-effective to manufacture and arrange many devices to work together. Not only are MEMS useful when implemented in micro-objects because of their ability to move freely in small spaces, they are also relatively inexpensive to produce using microelectronic fabrication techniques. Thin material layers are deposited as a base, patterned, and etched away, resulting in a microscopic 3-D structure. There are 2 categories of MEMS: sensors and actuators. Sensors retrieve information from surroundings, and actuators execute commands through controlled movements. Their small size and practical capabilities help to improve an extensive variety of technological products including hearing aids, accelerometers, insulin pumps, and inkjet printers [1]. The ability to quickly and accurately characterize the electro-mechanical parameters of a MEMS resonator is crucial to its effective application. Nonlinearities in device operation can result from various MEMS specifications such as excitation voltage, resonator structure, and device dimensions. Theoretical and numerical analysis on MEMS devices helps to accurately predict nonlinearities such as spring softening and hardening in the device. Spring softening is a reduction in the effective spring constant; it manifests as a downward shift in resonance frequency. Spring hardening is the opposite: an increase in the effective spring constant. The phenomena could occur simultaneously, but one usually dominates [2]. Analyzing the effects of nonlinearities will result in improved predictions of the actual behavior of resonators and similar
devices. Previous theoretical analysis of MEMS devices predicts spring softening and hardening effects. However, the theory and mathematical models have not yet been tested. Here, we present an experimental analysis of the nonlinearities in MEMS systems in accordance with the numerical methods previously presented by Elshurafa in [2]. Our experimental analysis will assist in confirming the accurate characterization of the electro-mechanical parameters of a MEMS resonator in the nonlinear regime. II. Device and Detection Scheme Figure 1: A diagram of a MEMS device used in this work. The plate and moveable electrodes are depicted by the lighter shaded regions, which are moveable. The darker shaded regions are fixed to the substrate and receive voltage excitations. The interleaving electrode teeth actuate the device, causing the center plate to oscillate through capacitive interactions [2]. Our MEMS device is a comb-drive micro-actuator that converts electrical excitation into a mechanical output using electrostatic forces. Knowledge of the geometry of the MEMS is crucial to understanding its operation. The device consists of a movable central plate suspended 0.75 µm above the substrate by four flexible serpentine springs [3]. Connected to the plate are six sets of comb electrodes, three on each side, which interleave with six sets of electrodes fixed to the substrate to form three sets of integrated parallel capacitors on each side with the plate as the common electrode [Fig. 1].
Figure 2: A close-up diagram of the plate and fixed electrodes on the MEMS device. The plate electrode is allowed to oscillate in the x-direction, while the fixed electrode is attached to the substrate and is stationary. Later, we will refer to the width of the combs in the y-direction as y. Referring to the z-axis depth of the electrodes as z, we treat the comb teeth as pairs of parallel capacitors, each with capacitances of C = εε 0A, where ε and ε 0are the relative permittivity and the permittivity of free space, respectively, d is the distance between teeth, and A = zδx [3]. The pairs of interdigitated teeth can be treated as parallel plate capacitors with a d capacitance of C = εε 0A d (see Fig. 2). Because the capacitors formed by the teeth are in parallel, we add their capacitances to obtain the total capacitance. N+1 teeth creates N pairings, so C = N εε 0z d (x 0 + x) = β(x 0 + x), where β = N εε 0z. β is the transduction factor, which relates d the electrical parameters, such as voltage and current, to the mechanical parameters, such as force and displacement. x changes as the plate displaces from equilibrium, therefore, the capacitance will vary as a function of x [4]. Figure 3: The circuit used to actuate the MEMS device. The AC and DC voltage sources are necessary for MEMS oscillation. The lock-in amplifier demodulates the signal from the charge-sensitive amplifier. The circuit in Figure 3 will give us a better picture of how the device is actuated and
detected. When an AC voltage is applied to the MEMS, differential capacitive forces are created, causing the comb electrodes connected to the plate to oscillate parallel to the substrate [5]. The electrodes on the left side of the MEMS will receive an AC voltage of V L = V ac, while the electrodes attached to the right side of the MEMS have DC voltage of V R = V dc. The voltage between pairs of electrodes creates an electric field with energy U = 1 2 CV2. Since F = du dx, an attractive force will develop between the electrodes of F = 1 dc V2 = 1 2 dx 2 βv2 [3]. We apply the difference between VL and VR this equation to get: F = 1 2 β(v ac 2 V dc 2 ) (1) To oscillate the MEMS, we are only interested in the sinusoidal force on the MEMS device. The magnitude of the oscillatory drive force becomes: F = 1 2 βv ac 2 (2) Therefore, when we drive at half of the detected frequency, V ac = V 0 e iωt 2, F = 1 2 βv 0 2 e iωt. (3) III. Damped Linear and Nonlinear Oscillators The amplitude of the device displacement can be found by solving the equation of motion for a damped harmonic oscillator [3]. In the linear regime, MEMS forces can be modeled by comparing the device to a damped driven harmonic oscillator, where the forces applied counter to its motion are the spring force, -kx, and the linear damping force, -γv. Newton s second law of motion tells us that: ma = F γv kx (4) where F is the oscillatory driving force on the plate, γ is the damping coefficient, and k is the spring constant of the device. We make the substitution k = ω 0 2 m, where ω 0 is the resonance
frequency and m is the mass of the device [4]. Replacing a and v for separable differential equations of x and rearranging gives us: The solution of this equation takes the form: F m = d2 x dt 2 + γ dx m dt + ω 0 2 x x = A 0 e iωt (5) (6) where A 0 is the amplitude of oscillation and ω is the driving force frequency from (3). Substituting (6) into (5) results in: F 0 m eiωt = A 0 ( ω 2 i γ m ω + ω 0 2 ) (7) Solving for A 0 : A 0 = F 0 m 1 (ω 0 2 ω 2 ) i γ m ω (8) To obtain the amplitude of the in-phase and out of phase term, we separate the real and complex components of A 0 [4]: A 0 = F 0 m (ω 0 2 ω 2 ) + i γ m ω (ω 0 2 ω 2 ) 2 + ( γ m ω)2 (9) We can plug A 0 back into the equation for x to get an expression for the displacement of the device. The lock-in amplifier detects the right-side MEMS voltage amplitude after current flows through a charge sensitive amplifier, as shown in Figure 3. The charge amplifier s output voltage as a function of input charge can be written as: V = αq (10) Where α is the amplification factor. Since q R = C R V R, and C R = βδx, the detected voltage will
Detected Voltage (V) be: V = αβ(x 0 + Δx)V dc (11) As previously shown, x varies as a function of time. Substituting Δx for x and incorporating equations (2), (6) and (9) to find the detected voltage results in [3]: V = αβ2 2 V dc V ac (ω 2 0 ω 2 ) + i γ m ω 2m (ω 2 0 ω 2 ) 2 + ( γ eiωt ω)2 m (12) Since we graph detected voltage as a function of frequency and not time, we neglect the time dependent term at the end. Separating the real (X) and complex (Y) terms of the detected voltage: V X = αβ2 V dc V ac 2 m V Y = αβ2 V dc V ac 2 m (ω 0 2 ω 2 ) (ω 0 2 ω 2 ) 2 + ( γ m ω)2 γω m (ω 0 2 ω 2 ) 2 + ( γ m ω)2 (13) (14) V X and V Y are the in-phase and out-of-phase expressions for detected voltage of the charge sensitive amplifier. The theoretical outputs are plotted in Fig. 4. V X 4.00E-06 3.00E-06 2.00E-06 1.00E-06 0.00E+00-1.00E-06-2.00E-06-3.00E-06-4.00E-06 24500 24700 24900 25100 25300 25500 Frequency (Hz)
Detected Voltage (V) V Y 3.50E-06 3.00E-06 2.50E-06 2.00E-06 1.50E-06 1.00E-06 5.00E-07 0.00E+00 24500 24700 24900 25100 25300 25500 Frequency (Hz) Figure 4: Theoretical plots of the in-phase (V X ) and out-of-phase (V Y ) components of the detected voltage with 10V DC and.5v AC when the resonance frequency is 25 khz, γ = 10 8 Ns/m, α= 3x10 8 V/C, β=10-9 F/m, and m=10-10 kg. Spring softening and hardening arise from nonlinearities in the restoring force, meaning that another term must be added to equation (4) to determine resonance frequency shifts due to hardening and softening effects on the MEMS resonator. In order to derive expressions for MEMS oscillator motion in the nonlinear regime, the well-known duffing equation for nonlinear damped harmonic motion is used: m d2 x dx + γ dt2 dt + mω 0 2 x + k n x 3 = F (15) where k n models nonlinearities in the restoring force. We note that the duffing equation matches the damped equation for harmonic motion shown previously when k n = 0. This equation has previously been used to calculate an expression for the resonance frequency shift in a MEMS device: δ = 3 8 KA max 2 2α 1 V dc 2 (1.5A max 2 + 1) (16) where K = k nx 0 2 ω 0 2 m, α 1 = Nε 0A 2mω 0 2 x 0 3, and δ = ω ω 0 ω 0. ω 0 is the original resonance frequency, m is the mass of the device, x 0 is the equilibrium distance between the plate and moveable electrodes, N is
the number of comb teeth on the device, A is the transverse area per comb tooth, and A max is the maximum amplitude of the device oscillation, which has also previously been derived to be: dc where α 2 = 1 = ε 2 dx 0 A max = 4V ac V dc α 2 Q N 1 2 (z d + 1 π ln ([(y d + 1)2 1][ 2d y + 1]1+y d)) (17) (18) y is the y-axis width of the combs, and Q is the quality factor, a dimensionless parameter describing how under-damped an oscillator is [2]. A high quality factor corresponds to a lower rate of damping. A depiction of equation for the frequency shift is shown in Figure 5. Increased AC excitation voltages were mathematically analyzed in [2] to show an increase in spring hardening and softening effects in MEMS oscillators [Fig. 5(c)]. Softening of the MEMS device will occur when δ < 0 and the peak shifts to the left. At low AC excitations, the frequency shift expression has been rearranged to show that δ < 0 when K << α 8 1V 2 dc, resulting in spring softening. The peak will shift to the right when δ > 0, K 8 >> α 1V dc 2 and spring hardening occurs [Fig. 5(b)]. a)
b) (c) Figure 5: Graphs of excitation frequency vs. amplitude of oscillation using models for typical responses of a MEMS resonator at various parameters. The grey sections of the graphs will follow one of the dotted lines during softening or hardening, depending on whether the frequency is sweeping up or down, before returning to blue line behavior. (a) The resonance of a MEMS device in the linear regime. (b) A positive resonance frequency shift due to spring hardening behavior at low AC excitations with a low DC bias. (c) Response of the device with a high AC excitation voltage. Spring softening and hardening effects become more apparent [2]. At low enough AC excitations, spring softening and hardening effects are dependent on only V dc. High V dc values were mathematically modeled to show spring softening, while low V dc values should result in spring hardening [Fig 5b]. Higher AC excitations have been modeled to show increased nonlinearities in MEMS resonators, as shown in Figure 4c. IV. Experimental Methods To confirm the spring softening and hardening expressions previously derived, the circuit in Figure 3 was created to first measure device resonance before shifting parameters to model nonlinearities. An EG&G Instruments 7265 DSP Lock-in Amplifier was used as both an AC voltage source and a lock-in amplifier to detect the signal coming from our pre-amplifier, which has a 3x10 8 amplification factor. A 16 Bit Bipolar DAC connected to a 10MΩ resistor was used for the DC bias. To ensure minimal damping, our MEMS device was enclosed in a cell attached to a BOC Edwards RV3 vacuum pump, which brought the cell pressure down to 10 mtorr. The MEMS cell was also connected to a cryo pump, which is an aluminum tube filled with activated
charcoal and cooled to 77K in operation. Activated charcoal has an extremely large surface area to mass ratio. When it cools to 77K, gas molecules will be physically adsorbed on the large surface, and the pressure in the cell will drop further to 5 mtorr, which ensures minimal damping from viscosity [2]. By replacing the vacuum pump during device operation, the cryo pump also eliminates potential electric noise. We run frequency sweeps using Labview 2013, which communicates with the lock-in internal oscillator and DC voltage source to set the frequency, AC amplitude, phase shift, and DC bias while detecting the voltage to the lock-in amplifier. Files were created containing the parameters and data for frequency versus detected X and Y voltage amplitude. These files are currently under analysis for spring softening and hardening effects. Mathematical models predict that higher AC excitations increase nonlinearities in spring hardening and softening. Theoretical analysis also calculates spring hardening to occur at low DC bias voltages, and spring softening to occur at high DC bias voltages. Testing of these models is currently underway. Further investigation is required to more accurately model frequency shifts in the MEMS device and compare the fits of our experimental data with MEMS numerical analysis. IV. Conclusions MEMS devices are modeled to follow force equations for nonlinear damped harmonic oscillators. Experimental measurement of MEMS resonance at varying frequencies is expected to show the device hardening and softening with changing and AC and DC voltage parameters. Previous numerical analysis on MEMS resonators have modeled the frequency shift caused by nonlinearities to follow: δ = 3 8 KA max 2 2α 1 V dc 2 (1.5A max 2 + 1) (19) If our experimental data agrees with the theoretical predictions in [2], at low AC
excitations K << α 8 1V 2 bias during spring hardening, and K >> α 8 1V 2 bias during spring softening. Experiments will be conducted with changing AC and DC parameters of MEMS devices to test whether the detected resonance peak shift follows previous theoretical and numerical analysis. Experimental confirmation of the models for nonlinearities in MEMS devices will help to better analyze and model these micro-machines, improving the quality of the cutting-edge technology in which they are implemented. Acknowledgement This research is supported in part by NSF DMR-1461019. Special thanks to Dr. Yoonseok Lee for his constant support and expertise in MEMS, as well as his positive attitude throughout the experiment. Much appreciation to Terrence Edmonds, who taught me valuable experimental techniques with MEMS while allowing me to assist him in summer research. Thank you to Colin Barquist for his consistent help and guidance, especially during experimental difficulties. I would like to thank Dr. Selman Hershfield and the University of Florida as a whole for providing me with a valuable research experience this summer that will greatly prepare me with the skills to excel in my future physics endeavors. References [1] MEMS Industry Group. 2009, Feb 17. Introduction to MEMS "Micro-Electro-Mechanical System." Retrieved from https://www.youtube.com/watch?v=cnmk-sem0zi [2] A M Elshurafa et al., J. Micromechanic. Sys. 4, 20 (2011). [3] C S Barquist et al., Signal analysis and characterization of a micro-electro-mechanical oscillator for the study of quantum fluids.
[4] J Bauer, Undergraduate Honors Thesis, University of Florida (2014). [5] M. González, P. Zheng, E. Garcell, Y. Lee, and H. B. Chan, Rev. of Sci. Instrum. 84, 025003 (2013).